Introduction to Elliptic Curves and Modular Forms

****NOTE: In previous course ad, I said that "You need to understand the definition of the Galois group Gal(\overline{K}/K) and its action on \overline{K}. (It would be preferable if you know some more Galois theory.) "
HOWEVER, we now do NOT require Galois theory at all! You simply need to know the materials in Algebra II and Complex analysis.

****Topics****
This is an introductory course on elliptic curves and modular forms, which are two (heavily inter-related) fundamental objects in modern algebraic number theory.

---We will first introduce the language of algebraic varieties (which also serves as an elementary prelude to algebraic geometry). Then we apply these algebraic geometry tools to study elliptic curves. In particular, we treat in more detail elliptic curves over complex numbers.
---Next, we introduce modular forms, whose definition only requires very basic complex analysis. Then we study the finer properties of modular forms, and explain how they are used in some number theoretic problems.
--- In the end (depending on the time we have), I will give a survey about the very deep connections between these various objects, in the framework of the far-reaching Langlands program. In order to do so, I will introduce the p-adic numbers, and some basic representation theory.

****Exams****
There will not be exams.
There will be exercises. To pass the course, you need to score 50% on the total exercises. Also, there is the optional choice to work on small projects (see below).
There are no different grades, just passed or non-passed.

****HOMEWORK POLICY:****
You can discuss with people about the problems. But you must write the answers on your own. (Do not copy from other people.)
Late homework submissions will not be accepted. (If it is something very serious, e.g., being sick for an entire week, then I will assign you some other work.)
Partial credits will be given (so please write down whatever that you get.)
You can submit your homework to me before deadline, either during class meetings, or emails (in pdf). Once submitted, no change is allowed.

****Small Projects.****
Students can do small projects to get score points. In general, you learn something (related) that is not covered/proved in our course, and write a short essay explaining it, and (time permitting), present it in classroom. You can do collaborated projects (up to 3 people).
I will assign percentage points. For example, I can assign 20% (of the whole course score) to your finished projects (depending on interest, scope, difficulty etc. of your project). Topics of the projects will be agreeed mutually between students and me.

Hui Gao

Published, 2.3.2018 at 16:01

week 7 log. we continued with Section 5 of Serre's book, and finished until Thm 7 and Prop. 13 ( we did not discuss L-function). In the final lecture, I gave a talk on Langlands program (it attracted around 10 people).

Hui Gao

Published, 21.2.2018 at 17:05

week 6 log. We finished Section 3 of Serre's book (space of mod forms); we also looked back, to show that complex tori correspond to complex EC via the modular invariant j. Then we start to discuss Section 5 of Serre's book; we defined T(n) and stated Prop. 10 there.

Hui Gao

Published, 14.2.2018 at 17:10

week 5 log. We discussed how to use Weirestrass functions to relate complex tori with complex elliptic curves. Then we move to define the modular forms (using Serre's book), and we showed that Eisenstein series are modular forms.

Hui Gao

Published, 7.2.2018 at 17:25

week 4 log. We discussed the group law. Then we moved to complex tori, and I finished [DS, Section 1.3]. Then I reviewed some basic complex analysis (residue theorem and some corollaries). Then I proved in detail [DS, P36, Exercise 1.4.1], which are important properties about a general elliptic function with respect to a lattice.

Hui Gao

Published, 31.1.2018 at 18:03

week 3 log. I finished lecturing Section 2 and 3 of [Sil, Chap 1]. I then started to lecture some general theory of elliptic curves, using materials in [Sil, Chap3]. I omit discussing singularites (node, cusps), and differential forms. In the end, I stated the group law.

Hui Gao

Published, 24.1.2018 at 17:55

Week 2 log (no week 1 lecturing). We started the course covering Chapter 1 of [Sil]. We assume the field K=R (real number), and \overline{K}=C (complex number) in the book. So the Galois group G_{C/R} is the group with 2 elements, generated by the complex conjugation action. With this, you do not need to know Galois theory to understand Chapter 1. (yet the key essence of Chapter 1 will not be lost).

Section 1. Basically, I skipped the parts which require Galois group (you only need to understand them when K=R). I do not prove Prop. 1.7. (you need to understand Example 1.8)
Section 2. I lectured until Example 2.8.

Hui Gao

Published, 15.1.2018 at 16:55

****NOTE: In previous course ad, I said that "You need to understand the definition of the Galois group Gal(\overline{K}/K) and its action on \overline{K}. (It would be preferable if you know some more Galois theory.) "
HOWEVER, we now do NOT require Galois theory at all! You simply need to know the materials in Algebra II and Complex analysis.